Spectrum of L\'evy-Ornstein-Uhlenbeck semigroups on $\mathbb{R}^d$

We investigate the spectral properties of Markov semigroups associated with Ornstein-Uhlenbeck (OU) processes driven by L\'evy processes. These semigroups are generated by non-local, non-self-adjoint operators. In the special case where the driving L\'evy process is Brownian motion, one recovers the classical diffusion OU semigroup, whose spectral properties have been extensively studied over past few decades. Our main results establish that, under suitable conditions on the L\'evy process, the spectrum of the L\'evy-OU semigroup in the $L^p$-space weighted with the invariant distribution coincides with that of the diffusion OU semigroup. Furthermore, when the drift matrix $B$ is diagonalizable with real eigenvalues, we derive explicit formulas for eigenfunctions and co-eigenfunctions--an observation that, to the best of our knowledge, has not appeared in the literature. We also show that the multiplicities of the eigenvalues remain independent of the choice of the L\'evy process. A key ingredient in our approach is intertwining relationship: we prove that every L\'evy-OU semigroup is intertwined with a diffusion OU semigroup. Additionally, we examine the compactness properties of these semigroups and provide examples of non-compact L\'evy-OU semigroups.